Graded geometry in gauge theories and beyond

We study some graded geometric constructions appearing naturally in the context of gauge
theories. Inspired by a known relation of gauging with equivariant cohomology we generalize the latter notion to the case of arbitrary Q -manifolds introducing thus the concept
of equivariant Q -cohomology. Using this concept we describe a procedure for analysis of
gauge symmetries of given functionals as well as for constructing functionals (sigma models) invariant under an action of some gauge group.
As the main example of application of these constructions we consider the twisted Poisson sigma model. We obtain it by a gauging-type procedure of the action of an essentially
infinite dimensional group and describe its symmetries in terms of classical differential
geometry.
We comment on other possible applications of the described concept including the
analysis of supersymmetric gauge theories and higher structures.